Derive ordered and unordered selection counts and account for repeated symbols.
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Assigning captain and recorder from five people gives 5×4=20 ordered role assignments. Selecting an unordered pair gives only 10 teams, because each team appeared once in each role order. Dividing is valid because every team is counted exactly twice.
Predict
Predict the overcount factor for a k-person committee. An ordered selection has n(n−1)…(n−k+1) possibilities. Each unordered k-subset admits k! orders, so C(n,k)=n!/[k!(n−k)!]. The factorial 0!=1 makes the empty selection count correctly.
Experiment
List ordered pairs from three named people, then group the six pairs into three unordered pairs. Try the same grouping with repeated symbols allowed: the outcome with two identical symbols has fewer distinct orders. This shows why one uniform divisor cannot be assumed without checking it.
Understand
Worked repeated-letter count: LEVEL has five letters, with two Ls, two Es and one V. Temporarily label the repeated copies, producing 5! arrangements. Each visible word is represented by 2!2! relabelings, so there are 30 distinct words. The equal overcount is what licenses division.
Permutation formulas answer ordered selection questions, not every problem containing the word “arrange.” Circular seating, identical objects and forbidden positions introduce additional equivalences or constraints. First describe which outcomes count as different.
Invalid argument: divide an ordered count by k because a team has k members. There are k! possible orders, not generally k. For k=3 the correction is six.
Apply
Write a short sentence defining the counted objects before using a formula. Check endpoint cases k=0 and k=n. If an outcome is impossible because k>n without repetition, the count is zero rather than a factorial of a negative integer.
Continue
Depth: choosing nonnegative counts of k categories with a fixed total leads to stars-and-bars, a different bijection. More advanced combinatorics studies partitions and symmetry groups; the formulas here should not be stretched to those cases without a new justification.
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Assign president, secretary and treasurer to distinct people from 7. How many assignments?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Choose an unordered committee of 3 from 8 distinct people. How many?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
How many distinct arrangements of the letters in AAB are there?
For any n≥0, the number of 0-element subsets is:
Derive C(n,k)=n!/[k!(n−k)!] for 0≤k≤n by forgetting order.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Count arrangements of BANANA and justify the repeated-letter correction.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
